Question 1

Which of the following best describes the meaning of the limit $$\displaystyle \lim_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ becomes arbitrarily close to $$L.$$

Question 2

Evaluate the following limit: $$\displaystyle \lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

Question 3

Which of the following functions is NOT continuous at $$x = 0$$?

Accepted Answer

$$f(x) = \frac{1}{x}$$

Question 4

A ball is thrown vertically upward with an initial velocity of $$20 m/s. $$Its height after $$t$$ seconds is given by $$h(t) = 20t - 4.9t^2$. What is the instantaneous velocity of the ball at t = 2$ seconds?

Accepted Answer

Question 5

Which rule is used to find the derivative of the product of two functions?

Accepted Answer

Product Rule

Question 6

Find the derivative of $$f(x) = x^3$ + 2x.$$

Accepted Answer

$$3x^2 + 2$

Question 7

A company’s profit $$P(x) in $$dollars from selling $$x$$ units is given by $$P(x) = -2x^2$ + 40x - 100. $$What is the marginal profit when $$x = 5$$?

Accepted Answer

$$20$$

Question 8

Which of the following is the correct application of the Chain Rule for $$y = \sin(3x^2$)$$?

Accepted Answer

$$y' = \cos(3x^2$) \cdot 6x$$

Question 9

Suppose $$f(x)$$ has a local maximum at $$x = 2. $$Which of the following must be true?

Accepted Answer

$$f'(2) = 0$$

Question 10

A tank is being filled with water at a rate given by $$R(t) = 5 + 2t$$ liters per minute. How much water enters the tank between $$t = 0$$ and $$t = 3$$ minutes?

Accepted Answer

$$24$$ liters

Question 11

Evaluate the definite integral $$\displaystyle \$$int_0$$^2 (3x^2$ + 2)\,dx.$$

Accepted Answer

$$16$$

Question 12

Which of the following statements about the Fundamental Theorem of Calculus is TRUE?

Accepted Answer

It relates differentiation and integration as inverse processes.

Question 13

A medication is administered at a rate of $$C(t) = 10e$$^{-0.5t}$$ mg/hour. What is the total amount administered from t = 0$ to t = 2$ hours?

Accepted Answer

$$10(1 - e$$^{-1}$$) mg$$

Question 14

Which of the following best describes the meaning of the derivative $$f'(a) in a $$real-world context?

Accepted Answer

The instantaneous rate of change of $$f(x) at$$ $$x = a$$

Question 15

If $$f(x) = \ln(x)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{1}{x}$$

Question 16

A car’s position is given by $$s(t) = t^3$ - 6t^2$ + 9t (in $$meters) at time $$t (in $$seconds). When is the car at rest?

Accepted Answer

$$t = 1$$ and $$t = 3$$

Question 17

Which of the following integrals represents the area between $$y = x^2$ and y = x$ from x = 0$ to x = 1$?

Accepted Answer

$$\displaystyle \$$int_0$$^1 (x - x^2$)\,dx$$

Question 18

A chemical reaction’s rate is modeled by $$r(t) = 4t - t^2$ (mol/L/min). When does the reaction reach its maximum rate?

Accepted Answer

$$t = 2$$ min

Question 19

Which of the following is the correct antiderivative of $$f(x) = 6x^2$?

Accepted Answer

$$2x^3 + C$

Question 20

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$Which theorem guarantees that there exists $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}$$?

Accepted Answer

Mean Value Theorem

Calculus I Course Syllabus and Weekly Structure

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